Theorem xpdom2g 6927

Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
xpdom2g	|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Proof of Theorem xpdom2g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4689	. . . . 5 |- ( x = C -> ( x X. A ) = ( C X. A ) )
2		xpeq1 4689	. . . . 5 |- ( x = C -> ( x X. B ) = ( C X. B ) )
3	1, 2	breq12d 4057	. . . 4 |- ( x = C -> ( ( x X. A ) ~<_ ( x X. B ) <-> ( C X. A ) ~<_ ( C X. B ) ) )
4	3	imbi2d 230	. . 3 |- ( x = C -> ( ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) ) <-> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) ) )
5		vex 2775	. . . 4 |- x e. _V
6	5	xpdom2 6926	. . 3 |- ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) )
7	4, 6	vtoclg 2833	. 2 |- ( C e. V -> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) )
8	7	imp 124	1 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4044   X. cxp 4673   ~<_ cdom 6826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279  df-dom 6829

This theorem is referenced by:  xpdom1g  6928